Sundry Reflections on Computer Science

Sundry Reflections on Computer Science 

 April 14, 1997

 The subject I intend to explore has several facets. First is the automated generation of valid mathematical theorems. Second are investigations of just which theorems may be generated by the computer. Finally, we have the actual practice of generating such theorems or unresolved problems. I still have to develop the relevant definitions of the subject matter such as ‘unresolved problems’, ‘computability and unsolvability’ and other such terms central to the area under study. The field has a long history with its origins in the ancient discipline of mathematical logic. However, modern computing hardware capabilities are increasingly up to the task of putting theory into practice.

For thousands of years questions of what is logically possible and how that knowledge can be applied to outstanding problems have perplexed and encouraged mathematicians and scientists. Developments in theoretical mathematics have often pointed the way to later developments in the sciences and especially to the models used to analyze the data those sciences are concerned with. One such modeling application is seen in the automation of algorithms that determine the computability of a given problem or that attempt to automate the search for the solution to an outstanding problem. This is seen as particularly valuable in modeling other complex scientific problems and extensive and complicated software projects. Artificial intelligence and mathematical combinatorics are just two of the areas where computability research has played a role.

What follows are several examples of my present still developing knowledge of the achievements and day to day work of this research. One example of a long standing problem only recently solved by the electronic computer is Robin’s Theorem. I also know that complex iterative procedures involving binary data trees in computer science are in the province of the field. These data structures seem particularly well suited to the demands of solvability research and vice versa. At the University of Minnesota Professor John Baxter works in this area.

In investigating this subject further there are at least two questions that I hope to answer or at least gain greater insight into the possibility of an answer. First, what big project, goal or dream research problem or application do researchers in this area have that they want to see resolved or investigated further? Second, what is the actual practice involved in achieving this and the efforts underway to ensure that the area continues to resolve and gain insight into new problems and continues to merit research?